A320826 Expansion of x*(1 - 4*x)^(3/2)/(3*x - 1)^2.

0, 1, 0, -3, -14, -51, -168, -521, -1542, -4365, -11740, -29439, -65670, -112273, -28344, 1018689, 6961550, 34606929, 151831044, 623095683, 2453975622, 9402575805, 35339538912, 130994480547, 480676041954, 1750847208621, 6343667488692, 22899720430251, 82466180250590

Offset: 0

Author

Peter Luschny, Oct 22 2018

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A002421, A320825, A320827.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!(x*(1-4*x)^(3/2)/(1-3*x)^2)); // G. C. Greubel, Oct 27 2018

Maple

c := n -> (-4)^(n-1)*binomial(3/2, n-1):
h := n -> hypergeom([2, 1 - n], [7/2 - n], 3/4):
A320826 := n -> c(n)*h(n): seq(simplify(A320826(n)), n=0..28);

Mathematica

CoefficientList[Series[(x (1 -  4 x)^(3/2))/(3 x - 1)^2, {x, 0, 28}], x]

PARI

x='x+O('x^30); concat([0], Vec(x*(1-4*x)^(3/2)/(1-3*x)^2)) \\ G. C. Greubel, Oct 27 2018

Formula

a(n) = c(n)*h(n) where c(n) = Catalan(n)*(3*n*(n + 1))/(2*(2*n-5)*(2*n-3)*(2*n-1)) = (-4)^(n-1)*binomial(3/2, n-1) and h(n) = hypergeom([2, 1 - n], [7/2 - n], 3/4).

A320826(n) = A320825(n) - A320827(n).